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Given a reductive group $G$ and a parabolic subgroup $P\subset G$, withmaximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of ageneric $T$-orbit in $G/P$, and determine in combinatorial termswhen the toricvariety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way theclassification of smooth Fano generic closures given by Voskresenski\u{\i} andKlyachko. As an application, we apply the well known correspondence betweenGorenstein Fano toric varieties and reflexive polytopes in order to exhibitwhich reflexive polytopes correspond to generic closures -- this list includesthe reflexive root polytopes.

Gorenstein Fano Generic Torus Orbit closures in $G/P$